HAL Id: hal-00368691
https://hal.archives-ouvertes.fr/hal-00368691
Submitted on 17 Mar 2009
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Extended Kalman Filter for Oversampled Dynamical
Phase Offset Estimation
Jordi Vilà Valls, Jean-Marc Brossier, Laurent Ros
To cite this version:
Jordi Vilà Valls, Jean-Marc Brossier, Laurent Ros. Extended Kalman Filter for Oversampled
Dynam-ical Phase Offset Estimation. ICC 2009 - IEEE International Conference on Communications, Jun
2009, Dresden, Germany. 5 p. �hal-00368691�
Extended Kalman Filter for Oversampled
Dynamical Phase Offset Estimation
Jordi Vil`a Valls, Student Member, IEEE, Jean-Marc Brossier and Laurent Ros
GIPSA-lab, Departement Image Signal BP 46 - 38402 Saint Martin d’H`eres - FRANCE
E-mail: jordi.vilavalls@gipsa-lab.inpg.fr, jean-marc.brossier@gipsa-lab.inpg.fr, laurent.ros@gipsa-lab.inpg.fr
Abstract—In this paper, we present an application of the Extended Kalman Filter for the on-line estimation of a dynamical carrier phase offset. The novel approach implies deriving the filter in an oversampled scenario in a digital receiver. We consider a Brownian phase evolution in a Data Aided scenario. Our numerical results using a BOC shaping pulse show that using the oversampled signal for estimating the phase offset we can obtain better performances than using a classical synchronizer.
Index Terms—Phase estimation, Extended Kalman Filter, over-sampling, carrier synchronization, GALILEO, BOC.
I. INTRODUCTION
Synchronization is a fundamental part in modern digital receivers. A synchronizer has to estimate some parameters, such as carrier frequency, carrier phase and timing epoch. This knowledge is required to recover the signal of interest correctly. In this paper we focus our attention on the phase estimation problem. Many methods for estimating the phase introduced by an unknown channel have been proposed over the past decades, from Phase Locked Loops (PLL) to the most sophisticated signal processing techniques. The Kalman Filter (KF) [3], [4], presented in early 1960s is one of the mostly used techniques for parameter estimation in linear gaussian problems. We can find extensive discussion on the KF in [5] and [6]. When dealing with nonlinear filtering problems, the Extended Kalman Filter (EKF) approximates the problem to apply the KF solution. Some contributions show the use of EKF for carrier phase recovery and frequency tracking [9]– [12]. To our knowledge, the EKF has never been applied to oversampled phase estimation for Binary Offset Carrier (BOC) shaped signals.
In this contribution, we investigate the use of an Extended Kalman Filter for carrier estimation in a Data Aided (DA) scenario. We consider an oversampled signal model after re-ceiver matched filter, this implies having a coloured reception
noise. This scenario is standard in radio-localization from a satellite signal. In [7], we have shown the potential gain for phase estimation provided by the use of the fractionaly-spaced signal after matched filter, instead of the symbol time-spaced signal. This was done by deriving a closed-form expression of the on-line Bayesian Cram´er-Rao Bound (BCRB) for the oversampled dynamical phase estimation. Now, our goal is to propose an EKF based algorithm which can approach this bound. We have thus to jointly estimate the coloured noise and the phase offset because the EKF doesn’t take it into
account. In Section II, we set the signal model. In Section III, we derive the expressions of the filter in the oversampled phase estimation scenario. In Section IV we recall the BCRB for this estimation problem. Finally in Section V, the numerical results for the EKF resulting from BPSK transmission are presented and interpreted.
N otations: italic indicates a scalar quantity, as in a; bold-face indicates a vector quantity, as in a and capital boldbold-face indicates a matrix quantity as in A. The (k, l)th
entry of a matrix A is denoted [A]k,l. The matrix transpose and
self-adjoint operators are denoted by the superscripts T and H respectively as in AT and AH.ℜ(·), ℑ(·) and (·)∗are the real
part, the imaginary part and conjugate of a complex number or matrix, respectively. Ex denotes the expectation over x.
II. SIGNALMODEL
We propose the signal model for the transmission of a known sequence {am}m∈Z over an Additive White Gaussian
Noise (AWGN) channel affected by a carrier phase offset θ(t).
A. Oversampled Signal Model
1) Discrete-time general formulation: the received complex baseband signal after matched filtering is
y(t) = "( TX m amΠ(t − mT ) ) eiθ(t)+ n(t) # ∗ Π∗(−t) (1) where T,Π(t) and n(t) stands for the symbol period, shap-ing pulse and circular gaussian noise with a known bilateral power spectral density (psd) N0.
We define the filtered noise
b(t) = [n(t)] ∗ Π∗(−t) (2) We also defineg˜m(t) as ˜ gm(t) = T Z +∞ −∞ Π∗(−α)eiθ(t−α+mT )Π(t − α)dα (3) Then the received signal can be written as
y(t) =X
m
Hereafter we suppose a shaping pulse with support in[0, T ] and a slow varying phase evolution during a period T . In this case we can approximateg˜m(t) by
˜ gm(t) ≈ g(t)eiθ(t+(m+ 1 2)T ) (5) where g(t) = T Z 0 −T Π∗(−α)Π(t − α)dα (6) If the received signal is fractionally-spaced at tk = kTS+ τ ,
where S is an integer oversampling factor and τ a known offset from the optimum sampling instants (we suppose0 ≤ τ ≤ T
S), we have that y „ kT S + τ « =X m am˜gm „ kT S + τ − mT « + b „ kT S + τ « (7)
and from (eq. 5) we have that
y kT S + τ = eiθ(kTS+τ + T 2)Ak+ b kT S + τ (8) where Ak = X m amg kT S + τ − mT . (9)
We can finally write the received oversampled signal as yk= Akeiθk+ b′k (10)
where k refers to tk instants.
We can define the symbol index p= ⌊k
S⌋, or equivalently,
k= pS + s with s the sample inside the symbol period. We note that s= 0, · · · , S − 1.
Note that the noise b′k is coloured with variance σn2, where
σ2
n = N0 × g(0)T is the variance of the AWGN noise n(t)
measured in the noise equivalent bandwidth of the receiver filterΠ∗(−t).
2) Discrete-time re-formulation for the noise: theTS-spaced sequence of noise,{b′
k}k∈Z, is defined in the previous section
from an analog noise n(t). Our motivation now is to replace this time serie by another {bk}k∈Z with the same statistical
properties, but which can be obtained entirely by a discrete-time formulation. This will be useful for the final state-space model formulation. We can write that
b′k = Z T 0 n α+ kT S + τ Π∗(α) dα = S−1X j=0 Z (j+1)T S jT S n α+ kT S + τ Π∗(α) dα
We define Γ as the covariance matrix of the observation noise. If we have N measurements the matrix is N× N and depends on the oversampling factor S.
The random variables
Zk,j= Z (j+1)T S jT S n α+ kT S + τ Π∗(α) dα are zero-mean gaussian distributed. For a fixed k, Zk,j are
independent in j. Their variance is equal to E|Zk,j|2 = N0 Z (j+1)T S jT S |Π (α)|2dα
We define a zero-mean, unit variance, gaussian i.i.d se-quence nk and Πj = ( N0 Z (j+1)T S jT S |Π (α)|2dα )1 2
Hence, the noise samples b′k have the same statistical
properties than samples bk obtained by a TS-spaced filtering
of the time serie nk:
bk = S−1X
j=0
Πjnk−j−1 (11)
B. Phase-offset Evolution Model
In practice we must consider jitters introduced by clocks imperfections. To take it into account we consider a Brownian phase-offset evolution [13]
θk= θk−1+ wk k≥ 2 (12)
where wk is an i.i.d. zero-mean Gaussian noise with known
variance σw2
S . Here σ 2
w stands for the variance growth of the
phase noise in one symbol period. The N × N covariance matrix of the phase-offset evolution is Σ.
C. State-Space Model
When using an optimal filtering approach a state-space model formulation is needed. Moreover, as we want to take into account that the observation noise is not white, we must include it into the state evolution.
We consider a sliding vector
νk νk−1 · · · νk−S+1
T
over an i.i.d. noise nk,
the evolution of this vector can be written as
2 6 6 4 νk νk−1 .. . νk−S+1 3 7 7 5 = 2 6 6 4 0 · · · 0 1 0 · · · 0 0 . .. ... 1 0 3 7 7 5 2 6 6 4 νk−1 νk−2 .. . νk−S 3 7 7 5 + 2 6 6 4 nk 0 .. . 0 3 7 7 5 (13)
The coloured noise bk is
bk= [Π0· · · ΠS−1] νk−1 νk−2 .. . νk−S (14)
The state to be considered includes the phase-offset and the coloured noise: xk = ˆ θk bk νk · · · νk−S+1 ˜ T . The state evolution is xk= Mxk−1+ wk (15) where M= 2 6 6 6 6 6 6 6 6 4 1 0 0 · · · 0 0 0 Π0 ΠS−1 0 0 0 0 · · · 0 .. . ... 1 . .. 0 0 · · · 0 1 0 3 7 7 7 7 7 7 7 7 5 (16) and wk= wk 0 nk 0 · · · 0 T (17) The observation equation can be written as
yk = Akexp i 1 0 · · · 0 Txk + 0 1 0 · · · 0 Txk (18) We note that the state equation is linear and the observation equation depends non-linearly on the state. With this formula-tion we have no observaformula-tion noise because we have included it in the state.
III. EXTENDEDKALMANFILTER
In the sequel we introduce the well known EKF. We can find the general EKF expressions in [5]. Then we derive the expressions on the oversampled carrier phase estimation scenario.
We have a system described by the following state-space equations pair
xk+1 = fk(xk) + wk
yk = gk(xk) + vk (19)
where xk is the state vector, wk is a zero-mean white noise
with covariance matrix Qk, ykis the observation vector at time
k which is a partial and noisy observation of the state xk and
vk is the observation noise with covariance matrix Rk
Both noises wkand vk are supposed to be uncorrelated. The
functions fk(·) and gk(·) can be non-linear in a general case.
For Gaussian, linear state models, the KF gives the best Mean Square Error (MSE) estimation of the state xk from
observations up to time k.
We notebxk|m, the estimation of xk from the observations up
to time m,exk|m = xk−bxk|m, the estimation error and Pk|m=
Eexk|mex T k|m
, the covariance matrix of the estimation error. The EKF gives us the estimatorbxk|kin a recursive way. The
main idea in the EKF is to linearize the state-space equations at each iteration in order to transform the filtering problem into a usual Kalman one.
So we need to compute
∂fk(xk)
∂xk
; ∂gk(xk) ∂xk
A. EKF for Dynamical Phase-Offset Estimation
Here we consider the oversampled phase estimation sce-nario. So our state-space model is the one presented in Section II (eqs. 15,18).
As the state equation is linear we have directly that ∂fk−1(xk)
∂xk
= M
The phase noise covariance Q is independent from k and has only two non-zero elements, namely
[Q]1,1=σ 2 w S ; [Q]3,3= σ 2 n
As we introduced the coloured noise bk into the state, there
is no observation noise and the covariance matrix R is null. The observation equation is non-linear with the state, so we have to apply a linearization. We define
g=∂gk ` bxk|k−1 ´ ∂xk =h iAkeibθk|k−1 1 0 · · · 0 iT (20)
Replacing these expressions into the general EKF expres-sions ( [5]) leads to the oversampled algorithm:
8 > > > > > > > > > > > > < > > > > > > > > > > > > : Pk|k−1 = MPk−1|k−1MH+ Qk−1 bxk|k−1 = Mbxk−1|k−1 Kk = Pk|k−1g H˘ gPk|k−1g H¯−1 Pk|k = [I − Kkg] Pk|k−1 bxk|k = bxk|k−1+ Kk h yk− Akeibθk|k−1− bbk|k−1 i (21)
IV. BAYESIANCRAMER´ -RAOBOUND
In this section we recall the expression of the on-line Bayesian Cram´er-Rao Bound (BCRB) for an oversampled phase estimation problem [7]. This bound is particularly suited for problems where an a priori information is available [8].
The BCRB matrix can be written as
BCRB= {B}−1=BD+ BP −1
(22) We have to compute two terms. The first one, BD, represents the average information about θ brought by the observations y, BDk,l= 2ℜnA∗lAk Γ−1k,leΨo (23) where Ψ = −1 2 Σ−1k,k+Σ−1l,l− 2Σ−1k,l (24)
−1 0 1 2 −1 −0.5 0 0.5 1 × T BOC Shaping Pulse
× 1 /T −1 0 1 −1 −0.5 0 0.5 1 BOC autocorrelation × T × 1 /T
Fig. 1. BOC shaping functionΠ(t) and its autocorrelation g(t)
The second term, BP, represents the information available from the prior knowledge on θ,
BP = 1 σ2 w/S 0 B B B B B B B @ 1 −1 0 · · · 0 −1 2 −1 . .. ... 0 . .. . .. . .. 0 .. . −1 2 −1 0 · · · 0 −1 1 1 C C C C C C C A (25)
In the general case, the on-line BCRB associated to obser-vation vector y= [y1· · · yN] is equal to entry (N, N ) of the
BCRB matrix,[BCRB]N,N.
As we analyse the estimation problem in a DA scenario the bound depends on the transmitted sequence a. In this paper we suppose the transmission of a known sequence to analyse the performance of the proposed algorithm and the bound. We note that, contrary to [7] where the proposed bound was the minimum over a set of sequences, the BCRB is computed in this paper over the transmitted known sequence.
When S > 1 we have more than one sample/symbol. In this case we have different bounds depending on the position inside the symbol so the BCRB depends on the couple(S, s). If we are interested in the optimal estimation values we set s= 0. So the bound is obtained as
BCRB(S, s) = [BCRB(a)]N,N (26)
with N= (M −1)∗S +s+1. M corresponds to the number of transmitted symbols, so the length of the known sequence in our case.
V. DISCUSSION
In this section we show the behaviour of the EKF by considering different scenarios. We assume the transmission over an AWGN channel of a M-sequences of length 511 bits, generated using a Linear Feedback Shift Register (LFSR) with characterisic polynomial [1021]8 (octal representation). We
consider three oversampling factors (S = 1,2 and 4) and a BOC shaping pulse (see figure 1). BOC shaping pulse is used in Galileo positioning system.
In the figures presented we plot the Mean Square Error (MSE) obtained by simulation versus the Signal to Noise Ratio
−20 −10 0 10 20 30 40 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 EKF versus SNR MSE(d B) SNR(dB) BCRB S=1 BCRB S=2 BCRB S=4 EKF S=1 EKF S=2 EKF S=4
Fig. 2. EKF and BCRB versus the SNR for three different oversampling factors S= 1, 2 and 4, with a phase-noise variance σ2
w= 0.001 rad2. (SNR). The SNR corresponds to the Carrier to Noise Ratio (NC) at the input of the receiver. In our case, as shaping pulse and symbols ak are normalised (i.e σ2a = 1; g(0) = 1) this
ratio is simply NC = 1
σn2. We compute the MSE for the T
-spaced optimal estimation values corresponding to the s= 0 case for all oversampling factors S. We also give the on-line BCRB for each case as a reference.
Figure 2 and 3 superimpose versus the SNR, the on-line BCRB (see eq.(26)) and the EKF. For figure 2, we have a slow varying phase with variance σw2 = 0.001 rad2 and for figure
3, we have a phase with a fast evolution, σ2w = 0.01 rad2.
In both scenarios there’s no offset from the optimal sampling instants, τ = 0.
One can see that for S = 1 the performance of the EKF is the same as the theoretical result of the BCRB. For S = 2 the performance of the algorithm are slightly looser when comparing to the bound. For S = 4 we obtain the same or slightly better performance as in the S= 2 case. In this case if we compare the algorithm with the bound we can see that the performance poorer. The gain increases with the oversampling factor S and the interest of oversampling becomes clear at low SNR. The gain due to oversampling decreases as the SNR increases.
In figure 4 we analyse the EKF behaviour for a fixed SNR versus phase-noise variance. We present a scenario with a low SNR value, SNR= 0dB. Here we can still measure the gain given by the oversampling and the good performance of the algorithm. The gain obtained with the oversampling is greater at weak σw2. We also note that the performance of the algorithm
at weak phase noise variance is really close from the bound. At very high σw2 the performances become poorer compared
to the bound. This is probably because for high SNR, the modeling error in the EKF linear approximation (see eq. 20) is not neglictible with respect to the noise level.
Figure 5 superimposes versus the SNR, the on-line BCRB and the EKF for a slow varying phase evolution scenario, σ2w = 0.001, and a non-null offset τ =
T
8 for S = 1 and
S = 2. We show in the same figure the performance of the EKF for a null offset τ = 0 as a reference.
−20 −10 0 10 20 30 40 −50 −40 −30 −20 −10 0 10 MSE(d B) SNR(dB) EKF versus SNR BCRB S=1 BCRB S=2 BCRB S=4 EKF S=1 EKF S=2 EKF S=4
Fig. 3. EKF and BCRB versus the SNR for three different oversampling factors S= 1, 2 and 4, with a phase-noise variance σ2
w= 0.01 rad2. 10−4 10−3 10−2 10−1 −40 −35 −30 −25 −20 −15 −10 −5 0 MSE(d B) !w2 EKF versus ! w 2 BCRB S=1 BCRB S=2 BCRB S=4 EKF S=1 EKF S=2 EKF S=4
Fig. 4. EKF and BCRB versus the phase noise variance for three different oversampling factors S= 1, 2 and 4, SNR = 0dB.
−20 −10 0 10 20 30 40 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 EKF versus SNR SNR(dB) MSE(d B) BCRB S=1 !=T/8 BCRB S=2 !=T/8 EKF S=1 !=T/8 EKF S=2 !=T/8 EKF S=1 !=0 EKF S=2 !=0
Fig. 5. EKF and BCRB in presence of a non nul offset for the sampling instants for two different oversampling factors S= 1 and 2, SNR = 0dB.
We can see that the performances for both the bound and the algorithm are looser when having a non-null offset τ . In general the performances decrease when increasing τ (0 ≤ τ ≤ T
S). We also note that the gain between different
oversampling factors is greater at high SNR when having a non-null offset.
VI. CONCLUSION
In this contribution, we have presented the Extended Kalman Filter for a realistic dynamical carrier phase estimation in an oversampled scenario. We have presented numerical results using a time limited pulse as done in satellite position-ing systems. In such scenario, where the Shannon samplposition-ing theorem is not respected, we have shown the interest of using a fractionally-spaced method for phase estimation. The interest of using this algorithm with the oversampled signal becomes clear at low SNR. The results obtained with the EKF are close to the theoretical bound for slow and moderate varying phase evolutions.
ACKNOWLEDGEMENT
This work was supported in part by the French ANR (Agence Nationale de la Recherche), LURGA project.
REFERENCES
[1] H. L. Van Trees, Detection, Estimation and Modulation Theory. New York: Wiley, 1968, vol.1.
[2] S. M. Kay, Fundamentals of statistical signal processing: estimation
theory. Upper Saddle River, NJ, USA: Prentice Hall, Inc., 1993. [3] R. E. Kalman, “ A new approach to linear filtering and prediction
problems”, J. Basic Eng., Trans. ASME, Series D, vol. 82, No.1, pp. 35-45, 1960.
[4] R. E. Kalman and R. S. Bucy “ New results in linear filtering and prediction theory”,J. Basic Eng., Trans. ASME, Series D, vol. 83, No.3, pp. 95-108, 1961.
[5] B. D. O Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ, USA: Prentice Hall, 1979.
[6] A. H. Jazwinski, Stochastic Processes and Filtering Theory. Academic Press, Inc. , NY, 1970.
[7] J.Vil`a Valls, J. M. Brossier and L. Ros, “ On-line bayesian Cram´er-Rao bound for oversampled dynamical phase offset estimation”, IEEE ISCCSP
2008, Malta, 12-14 March 2008.
[8] S. Bay, C. Herzet, J. M. Brossier, J. P. Barbot, and B. Geller, “ Analytic and Asymptotic Analysis of Bayesian Cram´er-Rao Bound for Dynamical Phase Offset Estimation”, IEEE Trans. Signal Processing, Jan. 2008. [9] P. J. Kootsookos, “ An extended kalman filter for demodulation of
polynomial phase signals”, IEEE Signal Processing Letters, Vol. 5, No. 3, March 1998.
[10] B. F. La Scala, R. R. Bitmead “ Design of an extended kalman filter frequency tracker”, IEEE Signal Processing Letters, Vol. 5, No. 3, March 1998.
[11] A. Aghamohammadi, H. Meyr, G. Ascheid“ Adaptive synchronization and channel parameter estimation using an extended kalman filter”, IEEE
Trans. on Communications, Vol. 37, No. 11, Nov. 1989.
[12] L.-L. Cheng, Z.-G. Cao “GPS Carrier phase measurement method based on data transition detection in high dynamic circumstance”, Proc.
WCC-ICCT, Vol.2, 21-25 Aug. 2000.
[13] P. O. Amblard, J. M. Brossier and E. Moisan, “ Phase tracking: what do we gain from optimality?Particle filtering versus phase-locked loops”,
